Tri-Vertices & SU(2)’s

Tri-Vertices & SU(2)’s

Amihay Hanany

Noppadol Mekareeya

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Tuesday, May 17, 2011

Introduction

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Introduction

• N=2 gauge theories in 3+1 dimensions

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Introduction


• Vector Multiplet Moduli Space

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Introduction



• Hypermultiplet Moduli Space

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Introduction




• revisit - study a class made out of SU(2)’s

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Introduction




• revisit - study a class made out of SU(2)’s

• interesting class of HyperKahler manifolds

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Hypermultipletmoduli space

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• Most attention in the literature - V plet moduli space (Coulomb branch)

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• Receives quantum corrections

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• H plet moduli space is classical

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• H plet moduli space is classical

• argue it has many interesting features

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From Quivers to Skeleton Diagrams

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• N=2 Quiver - nodes connected by lines

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• each node - V plet

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• each line - H plet, bifundamental

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• each line - H plet, bifundamental

• extend to other matter?

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Tri-Vertices

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Tri-Vertices

• First - change notation

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Tri-Vertices


• Inspired by brane configurations, use lines for V plets and nodes for H plets

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Tri-Vertices


• Inspired by brane configurations, use lines for V plets and nodes for H plets

• restrict to 3-valent vertices and lines representing SU(2) gauge group

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Skeleton Diagrams

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Skeleton Diagrams

• Lines & 3-valent Vertices

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Skeleton Diagrams


• Each Line - V plet with SU(2) gauge group

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Skeleton Diagrams



• Each vertex - a tri-fundamental of SU(2)3

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Skeleton Diagrams




• 8 half hypermultiplets in (2,2,2) of SU(2)3

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Skeleton Diagrams





• Length of the line L ~ 1/g 2

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Skeleton Diagrams





• Length of the line L ~ 1/g 2

• infinite line - global SU(2) symmetry

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Example: 8 free 1/2 Hypers

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• global symmetry SU(2)3

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• global symmetry SU(2)3

• No gauge group

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Example: N=4 SU(2) gauge theory

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• 1 gauge group

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• 1 gauge group

• 2 adjoints - an N=2 H plet

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• 1 gauge group


• 2 singlets - decouple

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• 1 gauge group



• infinite line - SU(2) global symmetry

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• 1 gauge group



• infinite line - SU(2) global symmetry

• possible N=2 breaking mass term

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Example: SU(2) with 4 flavors

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• 1 gauge group

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• 1 gauge group

• SU(2)4 global symmetry

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Infinite class of SU(2) theories

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• For each such diagram write a unique Lagrangian in 3+1 dimensions with N=2 supersymmetry

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• For each such diagram write a unique Lagrangian in 3+1 dimensions with N=2 supersymmetry

• Feynmann diagram in phi cubed field theory but each diagram represents a unique Lagrangian

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Conformal Invariance

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• Each finite line has 2 ends

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• giving 8 fundamental half hypers charged under it

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• giving 8 fundamental half hypers charged under it

• beta function is 0

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An infinite class of N=2 SCFT’s

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Topology of skeleton diagrams

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• each diagram has g loops & e external legs



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• Number of gauge groups 3g-3+e



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• Number of matter fields 2g-2+e



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• Number of matter fields 2g-2+e

• global symmetry SU(2)e


HypermultipletModuli Space

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• at generic point gauge group is broken down to U(1)g

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• dimension of moduli space (quaternionic)

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• (2g-2+e) x 8 x 1/2 - [(3g-3+e)x3 - g] = e+1

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• (2g-2+e) x 8 x 1/2 - [(3g-3+e)x3 - g] = e+1

• No dependence on g

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• (2g-2+e) x 8 x 1/2 - [(3g-3+e)x3 - g] = e+1

• No dependence on g

• Kibble Branch

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Look for generic resultsdepending on g & enot on Lagrangian

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Chiral Operators on Kibble Branch

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• Write the theory in N=1 language

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• Many chiral operators

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• Count Chiral Operators on the Kibble branch

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• Count Chiral Operators on the Kibble branch

• Hilbert Series

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What is aHilbert Series?

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g({ti}) =∑

i1,...,ik

di1,...,iktii

1· · · tik

k

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g({ti}) =∑

i1,...,ik

di1,...,iktii

1· · · tik

k

di1,...,ik

i1, . . . , ik

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Fugacities

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Fugacities

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• t - keeps track of the dimension of operator


Fugacities

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• x - SU(2)


Fugacities

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• x - SU(2)

• for e SU(2)’s there are e x’s


Fugacities

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• x - SU(2)

• for e SU(2)’s there are e x’s

• HS(t, x1, x2, ..., xe)


Example: g=0, e=3

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Example: g=0, e=3

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Example: g=0, e=3

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1 + [1; 1; 1]t + ([2; 2; 2] + [2; 0; 0] + [0; 2; 0] + [0; 0; 2])t2


Example: g=0, e=3

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1 + [1; 1; 1]t + ([2; 2; 2] + [2; 0; 0] + [0; 2; 0] + [0; 0; 2])t2

Qa,b,c


Example: g=0, e=3

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1 + [1; 1; 1]t + ([2; 2; 2] + [2; 0; 0] + [0; 2; 0] + [0; 0; 2])t2

Qa,b,cQ(a,b,cQa′,b′,c′)


Example: g=0, e=3

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1 + [1; 1; 1]t + ([2; 2; 2] + [2; 0; 0] + [0; 2; 0] + [0; 0; 2])t2

Qa,b,cQa,b,cQa′,b′,c′εa,a′

εb,b′Q(a,b,cQa′,b′,c′)


Example: g=0, e=3

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1 + [1; 1; 1]t + ([2; 2; 2] + [2; 0; 0] + [0; 2; 0] + [0; 0; 2])t2

Qa,b,cQa,b,cQa′,b′,c′εa,a′

εb,b′Q(a,b,cQa′,b′,c′)


Plethystics

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Example: g=0, e=4

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Example: g=0, e=4

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• To compute the HS observe


Example: g=0, e=4

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• Higgs branch is the moduli space of 1 SO(8) instanton on R4


Example: g=0, e=4

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• HS was computed and gives


Example: g=0, e=4

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• HS was computed and gives

•


HS: g=0, e=4

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HS: g=0, e=4

• Next decompose irreps of SO(8) to SU(2)4

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HS: g=0, e=4

• Next decompose irreps of SO(8) to SU(2)4

• Use fugacity map

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HS: g=0, e=4

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Gluing Theories

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Gluing Theories

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• 2 theories with (g1, e1) & (g2, e2)


Gluing Theories

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• glue by identifying an infinite leg from each


Gluing Theories

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• turn it into finite line


Gluing Theories

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• gauging an SU(2) global symmetry


Gluing Theories

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• gauging an SU(2) global symmetry

• get the theory with (g1+g2, e1+e2-2)


Gluing 2 theories with(0,3) to form (0,4)

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permutation symmetry

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permutation symmetry

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• The result has S4 symmetry - permutation of external legs (global symmetries)


Example: g=1, e=1

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g=1, e=1 another method

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• two commuting adjoints



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• two commuting adjoints

• symmetric product of 2 C2 s


Example: g=1, e=2

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Example: g=2, e=0

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examples: g=3, e=0

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Duality statement

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Duality statement

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• physics depends on g & e and not on the particular choice of the Lagrangian


Duality statement

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• physics depends on g & e and not on the particular choice of the Lagrangian

• Many to 1 correspondence


general case: (g,e)

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any g, e=0

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any g, e=1

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g, e=1

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g, e=1

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• complete intersection


g, e=1

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• generated by 5 operators


g, e=1

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• triplet of dimension 2; doublet of dimension 2g-1


g, e=1

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• triplet of dimension 2; doublet of dimension 2g-1

• one relation of degree 4g


generators of the moduli space

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generators

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generators

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• 3e at dimension 2


generators

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• 2e at dimension 2g-2+e


generators

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• Indication of the complexity at high e


generators

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• HyperKahler moduli space


generators

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• of dimension e+1


generators

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• of dimension e+1

• with SU(2)e isometry


Summary

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Summary

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• Study SU(2)’s & matter in tri-fundamental


Summary

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• properties of H plet moduli space (Kibble branch)


Summary

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• Special class of HyperKahler manifolds with SU(2)e isometries


Summary

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• Multi-ality: Physics depends on (g,e)


Summary

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• Multi-ality: Physics depends on (g,e)

• Symmetry: symmetric group in e elements


Thank you!

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Tri-Vertices & SU(2)’s